Preheat the oven to 350 degrees F (175 degrees C). Line a 12-cup muffin tin with paper baking cups.
Stir flour, sugar, nutmeg, baking powder, baking soda, and salt together for cupcakes in a medium bowl. Beat 1/2 cup butter in a large bowl until creamy, about 1 minute. Add flour mixture all at once and beat until mixture is crumbly.
Add eggs; beat about 30 seconds. Scrape the sides and bottom of the bowl well and beat until light in color and a bit fluffy, about 1 minute. Scrape the bowl again. Gently beat in eggnog just until combined and batter is silky and smooth. Divide batter evenly among prepared muffin cups.
Bake in the preheated oven until tops of cupcakes spring back when touched lightly and a toothpick inserted into the centers comes out clean, 25 to 28 minutes. Cool in the tin for 5 minutes. Transfer to a wire rack and let cool completely, about 20 minutes.
Meanwhile, whisk sugar and flour for frosting in a 2-quart saucepan. Whisk in eggnog and nutmeg until no lumps remain. Turn heat to medium and heat, beating with the whisk until mixture comes to a low boil and thickens enough to coat a spoon. Remove from the heat and scrape mixture into a bowl. Cool completely, about 45 minutes.
Beat butter and salt in a large bowl with an electric mixer until creamy. Pour in the cooled eggnog mixture, beating until light and creamy. If frosting looks curdled or separated, keep beating. It will come together after a few minutes.
Spread or pipe frosting onto the cooled cupcakes.
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